Variable Seperable Solutions in PDEs

Does anyone know of a proof of why, in partial DEs, one can assume the existence of variable seperable solutions, then take the linear combination of all of them to be the general solution? Why can't there be any other funny solutions that fall outside the space spanned by these variable seperable ones?

There is no such proof- first because, as you stated it, it isn't true. Solutions of linear pdes can be written as linear combinations (but you have to allow infinite sums or even integrals)- that's a basic property of linearity. Second, there doesn't have to be a proof that solutions of linear pdes can be written as sums of products of functions of the individual variables because you can prove, in analysis, that large spaces of functions, whether they are solutions to pdes or not, can be written in that way. Analytic functions can be written as infinite sums of products of powers of the variables- The Taylor's series. Periodic functions can be written as infinite sums of sines and cosines- The Fourier series. Much more general functions can be written as integrals of products of functions of the individual variables- The Fourier Transform. That has nothing to do with pde as such. The proof would be, for a specific pde, that the solutions must be in the required function space. That's why functional analysis plays such a large roll in partial differential equations.

From what I understand about your explanation, that large spaces of functions can be written as a linear combination of variable seperable solutions can be understood by considering the Taylor's series expansions of the solutions. But just another query here - Taylor's series converges only for a certain range that depends on the point from which the functions are expanded (of x and t, for example) for many functions I suppose? How can I link these "piece-wise" solutions to a general solution over the whole real line or complex plane?